Space Surveillance using Star Trackers. Part I: Simulations

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1 AAS Space Surveillance using Star Trackers. Part I: Simulations Iohan Ettouati, Daniele Mortari, and Thomas Pollock Texas A&M University, College Station, Texas Abstract This paper presents the first part of a demonstration on How to use Star Trackers to perform Space Surveillance. This part shows how to simulate the image taken by a Star Tracker placed on-board a spacecraft orbiting the Earth and how to use it for space surveillance. The simulation has to be accurate enough and take into account effects such as smearing and variable reflectivity of the object. Once the image is created and the usual treatments are performed, it can go to estimation routines and one can have an accurate idea of whether an object is detectable and whether its orbit can be estimated. 1 Introduction Star trackers are, presently, the most accurate attitude sensors on-board spacecraft. Their main purpose is to accurately evaluate the spacecraft orientation with respect to the Earth Centered Inertial (ECI) reference frame, where the star directions listed in the on-board star catalog are, usually, provided. Spacecraft attitude is estimated using the invariance of the inter-star angles with respect to orthogonal transformations. In other words, inter-star angles appear the same, independently from the spacecraft attitude. However, star trackers can also observe flying satellites due to the spacecraft reflectivity (albedo). This fact - supported by the recent increase of space threats - yields to the idea of using the existing star trackers sensors to perform surveillance of space objects from space. Identification of reflecting flying satellites relies on the capability to discriminate moving star-like objects in the sensor Field-Of-View (FOV). Two distinct approaches have been devised to identify the presence of a moving object within the camera FOV. One approach detects the motion by looking at the change in inter-star angles between un-identified and Paper AAS of the 16th AAS/AIAA Space Flight Mechanics Meeting, January 22-26, 2006, Tampa, Florida Invariance of inter-star angles on translations is also a well approximated assumption within the solar system. 1

2 identified stars while another approach detects the presence of a moving object by a subsequent image subtraction technique. Once a point source moving object is detected, it can be tracked and its orbit can be estimated. Reference [1] highlights which are the orbital parameters that can be fast estimated and those requiring longer observations, and which are the observation conditions in order to achieve a reasonable accuracy. In general, space surveillance using star tracker requires a high level of technology and bring the question: What can we do and what can we detect? This question can be answered in a first approach by creating a software simulating the image taken by digital star trackers. This software has to be reliable enough and to take into account effects such as smearing and variable magnitude of the target. This simulation software is then validated by feeding the on-board data processing software [2] (acquisition, centroiding, star identification, and attitude and covariance estimation) and by feeding the orbit estimation software [1]. Several steps are necessary to simulate the image. First step is the definition of the observing camera: lens focal length f and Charge-Coupled Device (CCD) parameters: noise σ, pixel size (d h, d v ), number of pixels N, Time Of Integration (TOI) T. All these constant parameters are stored in a structure format. Second step is to create the orbits of both observing and observed spacecraft. This step, apparently simple, requires extensive computational effort due to the searing simulation, which require orbit propagation over the TOI (on several intermediate points), which can be as small as 0.05 sec. In other words, the time step for the orbit computation must be small enough compared to the TOI so that the smearing effect can be accurately simulated. Since this step requires substantial computation effort, a particular attention is given. The simulation is performed by defining the initial time, t 0, and the number of orbital periods, n. The image is then simulated accordingly with the visibility conditions. On a tracking operative mode, the visibility condition depends on the observer/observed distance. As the observed object become visible, the focal axis start tracking the observed spacecraft and the image simulation process begins. Within the TOI, position and attitude of both objects slightly change. This affect both observed position direction as well as the apparent magnitude. The observed apparent magnitude is primarily function of the observer, observed, and Sun positions geometry as well as on reflectivity and attitude of the observed object. We are currently investigating the apparent brightness of several possible shapes and materials some of them having very low reflectivity. Once the observed direction is computed over the TOI, then the observed star list is generated. Also for the observed stars the smearing effect, due to the observer attitude dynamics, is simulated. Gaussian distributions are adopted to simulate the energy distribution of the focal plane for both observed stars or reflecting objects. A typical image is shown Fig.1 where an observed spacecraft appears at the center. Once the image series are simulated, standard image processing is performed, starting from If they do not change then, most likely, the un-identified star is just a dim star, not included in the on-board star catalog. 2

3 Figure 1: Typical image acquisition and noise reduction, up to attitude estimation. In particular, acquisition and centroiding are here proposed to be accomplished using the Run Length Encode (RLE) [3] that makes use of recursive functions. RLE scans the image horizontally and keeps record of all the stripes/segments of lights. For each stripe, the data set consisting of row index, and initial and final column pixels, is stored as a vector. Using a recursive function all the connected (or adjacent) stripes forming a single observation (star or object) are merged to form a unique shape. In particular, the software developed uses a double recursivity, vertical and horizontal, to identify shapes such as semi-circles and spirals. Once acquisition is performed, centroiding, star identification, and attitude estimation, are performed. Two difficulties arose while developing the simulation software. The first one is, as explained above, the high computational time necessary for the orbit computation. This step can take up to hundred minutes for both orbits, depending on the size of their semi-major axis. Thus, the difficulty is to find a way to make this process faster. A fifth order approximation method of the F&G series has been adopted, for a given number of steps before using the exact solution for a point in order to correct the accumulated error. The number of approximated steps is chosen accordingly with an assigned position tolerance. An upper bound of the error can be obtained by computing both approximated and exact solutions on a very eccentric orbit. The software simulates the attitude stabilization (three-axis or spinning) of the observed object, the type of material (reflectivity), the type of orbits, and the range of dates. 3

4 2 Star Tracker Simulations 2.1 Object Apparent Magnitude The apparent magnitude of the reflecting object is a function of the geometry (positions of Earth, Sun, observed, and observer). We consider two distinct cases: a) when the object is fully illuminated by the Sun and b) when it is hidden behind the Earth as shown in Fig. 2. When the angle α, identifying the angle between the Earth-to-Sun and the Earth-to- Figure 2: Positions of the Sun and the object with respect to the Earth Observer directions, becomes greater than the maximum value α max, then the Observer is hidden by the Earth, that is, it is located inside the Earth s shading cone. Mathematically, the illuminating condition implies r 2 where R E is the Earth radius and R S the Sun-Earth distance. R S R E R S sin α R E cos α = r 2 (1) At a distance of 1 AU, the value of the solar intensity can be considered constant and equal to 1, 367 W/m 2, throughout the year. The Earth reflects approximately 30% of this power while the Moon reflectivity can be neglected. Earth orbit propagation in heliocentric coordinates allows us to evaluate the Sun position in Earth Centered Inertial (ECI) at any time. Figures 3 and 4, show an observed object (at center of the image) that is simulated for the direct Sun (r 2 > r 2) and the Earth shadow (r 2 < r 2) illumination cases, respectively. The absolute magnitude of non-star objects (such as planets, comets, asteroids, and reflecting objects) is defined as the apparent magnitude that the object would have if it were at 1 AU distance from both the Sun and the Earth and at a phase angle ξ = 0. In particular, the absolute magnitude M of an observed spherical object (e.g., a planet or a satellite) can be approximated by the following expression M = m Sun 5 log 10 ( r d 0 a where m Sun = is the Sun apparent magnitude, d 0 = 1 AU = Km, r is the observed body radius, and a is the body albedo (reflectivity). The apparent magnitude, 4 ) (2)

5 Figure 3: Object fully illuminated m, can be evaluated from the absolute magnitude, M, by [ ] d 2 m = M log bs d 2 bo 10 d 4 0 p (ξ) where d bo and d bs are the observed-observer and observed-sun distances, respectively, and where p (ξ) is the Phase Integral (integration of reflected light) that for a spherical body can be well approximated by p (ξ) = 2 [ (π ξ) cos ξ + sin ξ ] (4) 3π Figure 5 shows the Phase Integral for an ideal diffusing reflecting sphere, which is a reasonable first approximation for spacecraft of unknown shape. The phase angle can be evaluated using where d os is the observer-to-sun distance (see Fig. 6). (3) cos ξ = d 2 bs + d 2 bo d 2 os 2d bo (5) 2.2 From Magnitude to Energy When photons emitted from an object (star or spacecraft) reach the Focal Plane Array (FPA), they are converted into electrons, the number of which is proportional to a brightness 5

6 Figure 4: Object partially illuminated on the resulting image. The longest the time of integration is, the more energy the FPA receives and the higher the number of electrons. The conversion process from the apparent magnitude of an object to its energy level is influenced by the lens aperture area of the camera. The apparent brightness, b obj, of an object at the pupil of the lens is given by The energy of a single photon is given by b obj = aperture 1, 370 ( ) 26.7 m (6) E pho = c λ where = J sec, is the Planck s Constant, c = Km/sec is the light speed, and λ is the wave length. The number of photons hitting the FPA per second is given by The total number of photons converted into energy by the FPA is (7) N = b obj E pho. (8) N P = N T TOI ρ (9) In most instances, the aperture ares is approximately the area of the entrance pupil of the lens. The apparent brightness is how much energy is coming from the star per square meter per second, as measured just above Earth s atmosphere. 6

7 Phase Integral Phase Angle (deg) Figure 5: Phase Integral Figure 6: Geometry and phase angle where T TOI is the TOI and ρ is a parameter that takes into account the losses in the optics. Finally, the total energy corresponding to a single object is E = N P E pho (10) Once this energy reaches the FPA, it suffers other losses due to the Quantum Efficiency, QEFF and is converted in electrons. The number of electrons finally counted at the output of the FPA is given by N e = QE.F F.N P (11) N e is expressed in electrons per second counted by the FPA. Multiplying N e by T T OI, one obtains the total number of electrons counted for the image. Those electrons are then spread upon the FPA according to the light distribution model. This distribution is usually selected as Gaussian since it well approximates Airy functions. 7

8 Figure 7: Field of view of the Camera 2.3 Unperturbed Observed Star The field of view of the camera, as shown in Fig. 7 is given by ( ) ( ) Lx Ly θ x = 2 tan and θ y = 2 tan 2f 2f The software uses a star-catalogue containing the coordinates and magnitude of 7, 000 stars. The coordinates are given in the geocentric reference frame using normalized vector. As stars are considered to be at an infinite distant, the satellite s altitude can be neglected. Knowing the observing spacecraft attitude, one can transfer the stars position vectors from the geocentric reference frame to the CCD reference frame (12) b = C s r (13) where r and b are the star position vectors as defined in geocentric and in the CCD reference frame, respectively, and C s is the observing spacecraft attitude. Then, using the co-linearity equation b = 1 x2 + y 2 + f 2 where x and y are respectively the horizontal and vertical coordinates of the reflected star on the CCD and f is the focal length, one can easily deduce the position (x, y) of the star on the CCD plane. 8 x y f (14)

9 A star appears in the field of view of the camera if the following conditions are verified ( ) ( ) x y tan 1 θ x tan 1 θ y and m m thr (15) f f where m is the apparent magnitude and m thr is the CCD magnitude threshold. Once the surrounding stars are selected, their position is computed at each step and a Gaussian distribution is used to spread the energy corresponding to the star s magnitude on the CCD. 3 Smearing 3.1 Orbital Dynamics The assumption here is that the observing spacecraft is tracking the observed spacecraft. The adopted system of coordinates is the geocentric Earth Centered Inertial (ECI). The procedure is explained as follows: To generate the observed spacecraft s orbit, the six orbital elements have been randomly chosen. Then, the orbit of the observed spacecraft can be computed with a given time step and the observing spacecraft is then placed within a sphere of radius 8 Km around the starting point of the observed spacecraft r 01 = r ε (16) where the indices 1 and 2 represent the observing and observed spacecrafts, respectively, and ε is a normalized random unit-vector. The velocity is then generated as follows where α is a random number ranging from 0.1 to 0.1. procedure are shown in Fig. 8. v 01 = v 02 + α v 02 ε (17) The results obtained from this As mentioned above, the time step for the orbit determination has to be small compared to the TOI which is, for this experiment, of the order of 200 ms. This means that the time step separating two points on the orbits has to be of the order of 50 ms. Thus, even in Low Earth Orbit, the orbit generation consumes a lot of computation time. A solution to solve this is to use the fifth order approximation of the F&G series for a given number of points and use the analytical solution at regular intervals as shown on Fig. 9. The F&G solution is used the following way { r = F r0 + G r 0 r = F r 0 + Ġ r 0 (18) 9

10 1 Orbits of the two objects z x y Figure 8: Tracking orbits One can, from an exact starting point, compute the coefficients of the series to the fifth order, higher order terms becoming more complex where F 0 = 1 F 1 = 0 F 2 = 1 2 ε 0 F 3 = 1 2 ε 0λ 0 F 4 = 5 8 ε 0λ ε 0ψ ε2 0 F 5 = 7 8 ε 0λ ε 0ψ 0 λ ε2 0λ 0 G 0 = 0 G 1 = 1 G 2 = 0 G 3 = 1 6 ε 0 G 4 = 1 4 ε 0λ 0 G 5 = 3 8 ε 0λ ε 0ψ ε2 0 ε 0 = µ λ r0 3 0 = r 0 v 0 and ψ r0 2 0 = v 0 v 0 (20) r0 2 The number of points separating two exact calculations has to be wisely chosen, so that the error remains below a given tolerance. This number can be computed with the case of a most-eccentric orbit where the velocity variation is the highest, thus we obtained a maximum for the error. 10 (19)

11 3.2 Attitude Dynamics Figure 9: Use of the F &G approximation Assuming a tracking mode, at the beginning of the TOI, the focal axis is aligned toward the observed object. One can then easily compute the Direction Cosine Matrix (DCM) describing the observing spacecraft s attitude. But this attitude is time-varying. The principal axis and angle of this rotation are randomly chosen. Concerning the principal axis, the angular displacement α of this vector with respect to the focal axis is bounded between 0 and 90. The principal axis is then given by e S = 2 2 sin α sin α 2 cos α as expressed in the CCD reference frame. The principal axis φ is given by (21) φ = ω t (22) where ω is the angular velocity of this rotation and δt is the time. parameters, one can compute the Rotation matrix where Knowing these two R = I cos φ + (1 cos φ) e S e T S + ẽ S sin φ (23) ẽ S = 0 e 3 +e 2 +e 3 0 e 1 e 2 +e 1 0 (24) One can then deduce the new attitude of the observing spacecraft by performing the matrix product C k+1 = C k R k (25) where k is the step index. One can then compute the new position of the target. 11

12 4 Simulations 4.1 Acquisition Once the image is created, one has to perform its acquisition. One way of doing this is the Run length Encode. This method consists in scanning the image horizontally, recording all the stripes of pixels which level of energy is greater than the threshold, row by row. These stripes are recorded in a n 3 vector V 1 = The first column of this vector represents the index of the row the stripe belongs to whereas the second and third column represent respectively the beginning and ending pixels of the stripe. This method has the advantages to be fast and clean. Moreover, the image does not need to be stored. Once V 1 is created, there is no need of the full image. Thus, this method saves memory. Once the above vector is created, one needs to assemble the stripes belonging to a same spot. This is done by looking for overlapping pixels from one row to another. In order to make this process easier, another vector is created V 2 = (26) (27) The first column of this vector represents the index of a row on the picture. The second and third represent the indexes of this row in V 1. For instance, the first row on the picture is represented by the rows one to three in V 1. This vector allows the routine to know exactly where to look for while searching overlapping. We have four types of overlapping pixels as shown in Fig. 10 where i and j represent respectively the above and below rows, and d and f represent the beginning and ending pixels of the stripe. Thus, if one of the four above conditions is verified, there is overlapping. One of the difficulties of acquisition appears when several stripes on a row belong to the same spot, as shown in Fig. 11. We can see that V 1 contains three spots, with one in a horizontal bone-shape. In order to solve this problem, a square matrix is created from V 1 containing all the interconnections 12

13 Figure 10: Overlapping pixels between two rows Figure 11: Acquisition of a semi-circle shape between stripes Then starting from row one of this matrix, the routine looks for 1. This routine is recursive. This means that whenever it finds a 1, it calls itself and look for other 1 on the row and column corresponding to the one found in the first place. For example, the routine is first going to find the 1 on the 1 st row and 5 th column. Once this 1 is found, it is replaced by a 0 to avoid duplicities and the routine calls itself, looking for 1 on the 5 th column and 5 th rows. The routine calls itself as many times as necessary to find all the connections. The following (28) 13

14 matrix is created and reduced to the following vector (29) (30) containing only as many columns as the number of spots on the image. From this vector, one can gather all the energy belonging to a spot and perform centroiding. Several tests have been performed on this routine with a double recursivity and the function turns out to be robust and time efficient on the shapes represented Fig. 12 Figure 12: Acquisition of complex shapes 14

15 However, attention should be put on the fact that this acquisition method is not made to deal with extended shapes. In such a case, the number of stripes is so important that any existing computer would run out of memory. Conclusion In order to simulate accurately the picture taken by the on-board CCD, one needs to compute every single parameter that impacts the result. This requires a lot of computational time and great emphasis should be placed on this point. Alternative ways like the fifth order approximation of the F &G series can be use in order to avoid this issue. Once the image is created, it is converted into a bitmap picture and centroiding techniques are applied so that estimation can be performed based on the simulation. Experimental and theoretical results can then be compared to check the accuracy of the simulation. Acknowledgment The authors would like to acknowledge Nick Comb and the Spacecraft Technology Center for their contribution on this topic. References [1] Ossama Omar Abdelkhalik, Daniele Mortari, and John Lee Junkins, Space Surveillance with Star Trackers. Part II: Orbit Estimation, Paper AAS of the 2006 Space Flight Mechanics Meeting Conference, Tampa, Florida, January [2] Daniele Mortari, Star Navigation and Spacecraft Attitude Determination, AERO-689 graduate course notes. Department of Aerospace Engineering, Texas A&M University, [3] Daniele Mortari, Christian Bruccoleri, Serena La Rosa, and John Lee Junkins, CCD Data Processing Improvements, International Conference on Dynamics and Control of Systems and Structures in Space 2002, King College, Cambridge, England, July 14 18,